What is the integral of (8-x)/((x^2+1)^2) ?

The integral of (8-x)/((x^2+1)^2) is 4arctan(x)+(4x)/(x^2+1)+1/(2(x^2+1))+C

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integral of (8-x)/((x^2+1)^2)

\int \frac{8 - x}{( x ^{2} + 1 ) ^{2}} d x

4 arctan (x) + \frac{4 x}{x ^{2} + 1} + \frac{1}{2 ( x ^{2} + 1 )} + C

Solution steps

\int \frac{8 - x}{( x ^{2} + 1 ) ^{2}} d x

Apply Integration By Parts

: \frac{1}{2} (8 - x) (arctan (x) + \frac{x}{1 + x ^{2}}) - \int - \frac{1}{2} (arctan (x) + \frac{x}{1 + x ^{2}}) d x

= \frac{1}{2} (8 - x) (arctan (x) + \frac{x}{1 + x ^{2}}) - \int - \frac{1}{2} (arctan (x) + \frac{x}{1 + x ^{2}}) d x

\int - \frac{1}{2} (arctan (x) + \frac{x}{1 + x ^{2}}) d x = - \frac{1}{2} (x (arctan (x) + \frac{x}{1 + x ^{2}}) + \frac{1}{x ^{2} + 1})

= \frac{1}{2} (8 - x) (arctan (x) + \frac{x}{1 + x ^{2}}) - (- \frac{1}{2} (x (arctan (x) + \frac{x}{1 + x ^{2}}) + \frac{1}{x ^{2} + 1}))

Simplify

\frac{1}{2} (8 - x) (arctan (x) + \frac{x}{1 + x ^{2}}) - (- \frac{1}{2} (x (arctan (x) + \frac{x}{1 + x ^{2}}) + \frac{1}{x ^{2} + 1})) : 4 arctan (x) + \frac{4 x}{x ^{2} + 1} + \frac{1}{2 ( x ^{2} + 1 )}

= 4 arctan (x) + \frac{4 x}{x ^{2} + 1} + \frac{1}{2 ( x ^{2} + 1 )}

Add a constant to the solution

= 4 arctan (x) + \frac{4 x}{x ^{2} + 1} + \frac{1}{2 ( x ^{2} + 1 )} + C

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What is the integral of (8-x)/((x^2+1)^2) ?
The integral of (8-x)/((x^2+1)^2) is 4arctan(x)+(4x)/(x^2+1)+1/(2(x^2+1))+C